Justin Welby, the bishop of Durham, has urged the Church of England's General Synod to vote in favour of legislation allowing women to become bishops as his appointment as the next archbishop of Canterbury was confirmed by Downing Street.

Speaking at his inaugural press conference as successor to Rowan Williams, the 56-year-old said he was "overwhelmed and surprised" to discover he had been offered the job of leader of the Church of England and figurehead of the Anglican communion.

While he said he was "utterly optimistic" about the church's future capacity for growth, he acknowledged that it faced challenges on the divisive issues of gay and women's rights. The synod, the church's parliament, is to vote later this month on whether to allow women to become bishops, and campaigners for the change say they fear they may lose.

"I will be voting in favour, and join my voice to many others in urging the synod to go forward with this change," said Welby, in a strong endorsement of their position.

Admitting that the church faced "deep differences" on sexuality, Welby, an evangelical, said he would not tolerate prejudice. But he said he supported his fellow bishops in their opposition to government plans to legalise gay marriage. "It is absolutely right for the state to define the rights and status of people cohabiting in different forms of relationships, including civil partnerships. We must have no truck with any form of homophobia, in any part of the church," he said.

While it was important for the church to shoulder its responsibility to other countries in the Anglican communion such as Nigeria, he added, it was also essential that as archbishop he listened carefully to the point of view of gay believers. "I also know I need to listen very attentively to the LGBT [lesbian, gay, bisexual, and transgender] communities and examine my own thinking carefully and prayerfully," he said. "I am always averse to the language of exclusion, where what we are called to is to love in the same way that Jesus Christ loves us. Above all, in the church we need to create safe spaces for these issues to be discussed in honesty and in love."

Welby, who was enthroned as a bishop last November, presented a jocular, relaxed face to the press as he appeared for the first time at Lambeth Palace, surrounded by the portraits of archbishops past. He even said he was eager to convey thoughts from the throne of St Augustine via the medium of Twitter.

Paying tribute to his immediate predecessor as a man of "infectious spirituality" who would one day be recognised "as one of the greatest archbishops of Canterbury", he said: "On the basis that you should only follow failures, this is a great mistake."

Williams, who steps down at the end of the year, returned the compliment, saying he was "delighted" at the appointment.

"He has an extraordinary range of skills and is a person of grace, patience, wisdom and humour," he said. "He will bring to this office both a rich pastoral experience and a keen sense of international priorities, for church and world."

David Cameron, whose office formally announced the appointment on Friday morning, said Welby had been the "overwhelming choice" of the Crown Nominations Commission, a body of clergy and lay people. He added: "I think having someone who had a life outside the church in business, who understands difficult, complicated issues, will bring a great breath of fresh air to the Church of England."

Welby said that he had found out about his new job while rushing to a meeting, and had opened the letter from Cameron on the street. His immediate reaction, he said, was: "Oh, no."

"I was chasing off to something last Friday, trying to put bits and pieces together for a meeting, most of which I've lost, and the phone went and I muttered something rude about phones and answered it, and the prime minister's appointment secretary said, 'We are at the moment writing to you; the prime minister is writing to you to offer you the see of Canterbury and the letter will get to you on Monday', which it duly did. It was handed to me outside Portcullis House in the street, so that was quite amusing.'

"I opened it then and there, I'm afraid – never good at waiting for things. My initial reaction, I have to be absolutely honest, when he said what he said on the phone, my initial reaction was 'Oh, no', but I was just overwhelmed and surprised by it."

Welby, an Old Etonian who worked in the oil industry for 11 years, will keep his place on the parliamentary inquiry into banking ethics. At Lambeth Palace he rejected any suggestion that he was somehow defined by his education, saying: "It's a very long time ago, sadly, that I left school … I think, I hope, I am defined because I love and follow Jesus Christ."

Welby's appointment received widespread support from other bishops. The archbishop of York, John Sentamu, also took the opportunity to hit out at the secretive appointment process, saying: "The appointment of an archbishop is neither akin to a horse race nor a presidential campaign, and it is a relief that the rumour mill, which has been grinding out misinformation, has now ground to a halt."

Welby made a joke of the furious speculation, quipping: "This is the best-kept secret since the last cabinet reshuffle."

He will be enthroned as archbishop of Canterbury in Canterbury Cathedral on 21 March.